Operations on MOSes: Difference between revisions

Line 1:

== Sistering ==

'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an r-step scale pattern ~~a_1 X_1~~... ~~a_r X_r~~ we ~~could~~ call the set of all patterns ~~a_1 X_pi~~(1) ... ~~a_r X_pi~~(r) over all permutations pi on {1, ..., r} the ''sisterhood'' of ~~a_1 X_1~~... ~~a_r X_r~~.)

'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an ''r''-step scale pattern ''a1X1 ... arXr'' with ''r'' step sizes ''X''1 > ... > ''Xr'', we call the set of all patterns ''aπ(1)X1 ... aπ(r)Xr'' over all permutations π on {1, ..., ''r''} the ''sisterhood'' of ''a1 X1 ... arXr''.)

If xL ys has a generator range between a\x and b\(x+y) (it always holds that a < b), then its sister yL xs has a generator range between b\(x+y) and (b-a)\y.

@@ Line 1: / Line 1: @@
 == Sistering ==
-'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an r-step scale pattern a_1 X_1... a_r X_r we could call the set of all patterns a_1 X_pi(1) ... a_r X_pi(r) over all permutations pi on {1, ..., r} the ''sisterhood'' of a_1 X_1... a_r X_r.)
+'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an ''r''-step scale pattern ''a<sub>1</sub>X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub>'' with ''r'' step sizes ''X''<sub>1</sub> > ... > ''X<sub>r</sub>'', we call the set of all patterns ''a<sub>π(1)</sub>X<sub>1</sub> ... a<sub>π(r)</sub>X<sub>r</sub>'' over all permutations π on {1, ..., ''r''} the ''sisterhood'' of ''a<sub>1</sub> X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub>''.)
 If xL ys has a generator range between a\x and b\(x+y) (it always holds that a < b), then its sister yL xs has a generator range between b\(x+y) and (b-a)\y.